What do we mean by 'Equivalent Projective representation"?


by S_klogW
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S_klogW
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#1
Dec18-12, 11:04 AM
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I know that we say two representations R and R' of a group G is equivalent if there exists a unitary matrix U such that URU^(-1)=R'.
But what do we mean by equivalent projective rerpesentations?
I've heard of the theorem that the SO(3) group has only 2 inequivalent projective representations. But what does that exactly mean?
I am very interested in projective representation because it's projective representation rather than ordinary representation that represents symmetry in Quantum Mechanics since the vector A and exp(id)A represent the same physical state.
So does anyone know if there are some books that can serve as an introduction to projective representations?
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Fredrik
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Dec21-12, 06:27 PM
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Since you haven't received any replies, I will mention that "Geometry of quantum theory" by Varadarajan covers projective representations and their relevance to quantum mechanics. I hesitate to recommend it because I find it very hard to read, but I don't know a better option.
micromass
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Dec21-12, 07:51 PM
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I've never done anything about projective representations before, so this post is just a guess. But it would make sense to define first

[tex]Z=\{cI_n~\vert~c\in \mathbb{R}\}[/tex]

Then we define a projective representation as a group homomorphism

[tex]\rho: G\rightarrow GL_n(\mathbb{R})/Z[/tex]

This last group is often called [itex]PGL_n(\mathbb{R})[/itex], or the projective general linear group.

Given, [itex]\rho,\rho^\prime[/itex] projective representations, it would make sense to define them equivalent if there exist [itex]U\in O_n(\mathbb{R})/Z[/itex] such that

[tex]\rho(g)=U\cdot \rho^\prime(g)\cdot U^{-1}[/tex]

for all [itex]g\in G[/itex].

The complex case is similar.


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